Ordering near the percolation threshold in models of 2D interacting bosons with quenched dilution

TL;DR

This paper investigates how quantum bosons with quenched dilution in 2D can exhibit ordering near the percolation threshold, combining percolation theory with quantum lattice models, and introduces a new RSRG fixed point supporting long-range order.

Contribution

It provides an analytic study of 2D quantum boson models near percolation threshold, revealing a new fixed point that supports long-range order despite fractal cluster geometry.

Findings

Spin-wave theory predicts long-range order but misses rare cluster points.

Real-space RG identifies fixed points supporting order on the percolation backbone.

Results are relevant for experiments with optical lattices and superconducting arrays.

Abstract

Randomly diluted quantum boson and spin models in two dimensions combine the physics of classical percolation with the well-known dimensionality dependence of ordering in quantum lattice models. This combination is rather subtle for models that order in two dimensions but have no true order in one dimension, as the percolation cluster near threshold is a fractal of dimension between 1 and 2: two experimentally relevant examples are the O(2) quantum rotor and the Heisenberg antiferromagnet. We study two analytic descriptions of the O(2) quantum rotor near the percolation threshold. First a spin-wave expansion is shown to predict long-ranged order, but there are statistically rare points on the cluster that violate the standard assumptions of spin-wave theory. A real-space renormalization group (RSRG) approach is then used to understand how these rare points modify ordering of the O(2) rotor. A new class of fixed points of the RSRG equations for disordered 1D bosons is identified and shown to support the existence of long-range order on the percolation backbone in two dimensions. These results are relevant to experiments on bosons in optical lattices and superconducting arrays, and also (qualitatively) for the diluted Heisenberg antiferromagnet.